whether the best way to present more of the backstory is in a frame or as an independent story

I think that's difficult to judge when you're still working on the story. Depending on how you structure it, a place may open up for that in the evolving story, especially if it gives us an insight into Illya that we need. At the same time, if the backstory starts to take over completely, you may well want to just write a separate story for it. How unhelpful is that :)
I'll read either though.